computational complexity

computational complexity, a measure of the amount of computing resources (time and space) that a particular algorithm consumes when it runs. Computer scientists use mathematical measures of complexity that allow them to predict, before writing the code, how fast an algorithm will run and how much memory it will require. Such predictions are important guides for programmers implementing and selecting algorithms for real-world applications.

Computational complexity is a continuum, in that some algorithms require linear time (that is, the time required increases directly with the number of items or nodes in the list, graph, or network being processed), whereas others require quadratic or even exponential time to complete (that is, the time required increases with the number of items squared or with the exponential of that number). At the far end of this continuum lie intractable problems—those whose solutions cannot be efficiently implemented. For those problems, computer scientists seek to find heuristic algorithms that can almost solve the problem and run in a reasonable amount of time.

Further away still are those algorithmic problems that can be stated but are not solvable; that is, one can prove that no program can be written to solve the problem. A classic example of an unsolvable algorithmic problem is the halting problem, which states that no program can be written that can predict whether or not any other program halts after a finite number of steps. The unsolvability of the halting problem has immediate practical bearing on software development. For instance, it would be frivolous to try to develop a software tool that predicts whether another program being developed has an infinite loop in it (although having such a tool would be immensely beneficial).